Discrete Compactness for the p-Version of Discrete Differential Forms

Abstract

In this paper we prove the discrete compactness property for a wide class of p finite element approximations of non-elliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find sufficient conditions for the p-version of a generalized discrete compactness property, which is formulated in the setting of discrete differential forms of order l on a polyhedral domain in Rd (0 < l < d). One of the main tools for the analysis is a recently introduced smoothed Poincaré lifting operator [M. Costabel and A. McIntosh, On Bogovskiı̆ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2009)]. In the case l = 1 our analysis shows that several widely used families of edge finite elements satisfy the discrete compactness property in p and hence provide convergent solutions to the Maxwell eigenvalue problem. In particular, Nédélec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind) are covered by our theory. Dipartimento di Matematica “F. Casorati”, Università di Pavia, I-27100 Pavia, Italy, daniele.boffi@unipv.it IRMAR, Institut Mathématique, Université de Rennes 1, 35042 Rennes, France, martin.costabel@univ-rennes1.fr IRMAR, Institut Mathématique, Université de Rennes 1, 35042 Rennes, France, monique.dauge@univ-rennes1.fr The Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA, leszek@ices.utexas.edu SAM, ETH Zürich, CH-8092 Zürich, hiptmair@sam.math.ethz.ch

DOI: 10.1137/090772629

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Cite this paper

@article{Boffi2011DiscreteCF, title={Discrete Compactness for the p-Version of Discrete Differential Forms}, author={Daniele Boffi and Martin Costabel and Monique Dauge and Leszek F. Demkowicz and Ralf Hiptmair}, journal={SIAM J. Numerical Analysis}, year={2011}, volume={49}, pages={135-158} }